This textbook is for the standard, one-semester, junior-senior course that often goes by the title “Elementary Partial Differential Equations” or “Boundary Value Problems” The audience usually consists of students in mathematics, engineering, and the physical sciences. The topics include derivations of some of the standard equations of mathematical physics (including the heat equation, the wave equation, and the Laplace’s equation) and methods for solving those equations on bounded and unbounded domains. Methods include eigenfunction expansions or separation of variables, and methods based on Fourier and Laplace transforms. Prerequisites include calculus and a post-calculus differential equations course.
There are several excellent texts for this course, so one can legitimately ask why one would wish to write another. A survey of the content of the existing titles shows that their scope is broad and the analysis detailed; and they often exceed five hundred pages in length. These books generally have enough material for two, three, or even four semesters. Yet, many undergraduate courses are one-semester courses. The author has often felt that students become a little uncomfortable when an instructor jumps around in a long volume searching for the right topics, or only partially covers some topics; but they are secure in completely mastering a short, well-defined introduction. This text was written to provide a brief, one-semester introduction to partial differential equations. It is limited in both scope and depth compared with existing books, yet it covers the main topics usually studied in the standard course and also provides an introduction to using computer algebra packages to solve and understand partial differential equations. The frontiers of mathematics and science are receding rapidly, and a one-semester course must try to advance the students to a level where they can reach these boundaries more quickly than in the past. Not every traditional topic can be covered, and not every topic can be examined in great detail. An example is the method of separation of variables, which plays a dominant role in most texts. It is this author’s view that a few well-chosen illustrations of the method of separation of variables should suffice.
The level of exposition in this text is slightly higher than one usually encounters in the post-calculus differential equations course. The philosophy here is that a student should progress in his or her ability to read mathematics. Elementary calculus texts, and even ordinary differential equations texts, contain lots of examples and detailed calculations, but advanced mathematics and science books leave a lot to the reader. This text leaves some of the details that are easy to supply to the reader. The student is encouraged as part of the learning process to fill in these missing details (see “To the Student”). The writing has more of an engineering and science style to it than a traditional, mathematical, theorem-proof format. Consequently, the arguments given are “derivations” in lieu of carefully constructed proofs.
Tb the Student
Chapter 1: The Physical Origins of Partial Differential Equations
Chapter 2: Partial Differential Equations on Unbounded Domains
Chapter 3: Orthogonal Expansions
Chapter 4: Partial Differential Equations on Bounded Domains
Appendix: Ordinary Differential Equations
Table of Laplace Transforms
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